I watched a TV show with Japanese subtitles, and noticed the sentence:
"秘密にしててほんとごめん。"
I was a bit confused because didn't know why there was a second て after the te-form of する. Because I didn't know how to look this up in my textbook or on Google, I asked a certain AI chatbot about it.
It tells me that してて is actually a contraction of していて (te-form of する and いる).
秘密にしててほんとごめん。 meaning "I'm really sorry for keeping it a secret."
秘密にしてほんとごめん。(without the second て) would only mean "I'm really sorry for making it a secret."
Yes, it's correct. It's not uncommon to have shortcuts in Japanese, especially if the connecting sounds are phonetically similar-ish. Casual masculine Japanese is particularly guilty of shortening up words and phrases, but it's common for both sexes.
Yes. The term you're looking for is い抜き. It's one of those things that's technically, grammatically wrong and improper, but has been getting more and more common in casual speech, along with dropping "ra" or needlessly adding "re".
ゆっくりしていってね!Japanese uses both して and していって (can sound like してって). Adding いって to something gives it a sense of being "ongoing" and other forms of て+いる also exist